Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910201979863040 |
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| author | Elliott, Chris Murfet, Daniel |
| author_facet | Elliott, Chris Murfet, Daniel |
| contents | These notes introduce the theory of susceptibilities as developed in [arXiv:2504.18274, arXiv:2601.12703] for interpreting neural networks. The susceptibility of an observable $ϕ$ to a data perturbation is defined as a derivative of a posterior expectation, which by the fluctuation--dissipation theorem equals a posterior covariance. Different choices of $ϕ$ yield different objects: per-sample losses give the influence matrix (the Bayesian influence function of [arXiv:2509.26544]), while component-localized observables give the structural susceptibility matrix that pairs model components with data patterns. The susceptibility matrix is (up to a factor of $nβ$) the Jacobian of the map from data distributions to structural coordinates; its pseudo-inverse provides a linearized solution to the patterning problem of [arXiv:2601.13548]: finding data perturbations that produce a desired structural change. We motivate the theory from its statistical-mechanical foundations, then give a detailed exposition of susceptibilities, their empirical estimators, and their connection to the geometry of the loss landscape. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_07980 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning Elliott, Chris Murfet, Daniel Machine Learning Statistical Mechanics Statistics Theory These notes introduce the theory of susceptibilities as developed in [arXiv:2504.18274, arXiv:2601.12703] for interpreting neural networks. The susceptibility of an observable $ϕ$ to a data perturbation is defined as a derivative of a posterior expectation, which by the fluctuation--dissipation theorem equals a posterior covariance. Different choices of $ϕ$ yield different objects: per-sample losses give the influence matrix (the Bayesian influence function of [arXiv:2509.26544]), while component-localized observables give the structural susceptibility matrix that pairs model components with data patterns. The susceptibility matrix is (up to a factor of $nβ$) the Jacobian of the map from data distributions to structural coordinates; its pseudo-inverse provides a linearized solution to the patterning problem of [arXiv:2601.13548]: finding data perturbations that produce a desired structural change. We motivate the theory from its statistical-mechanical foundations, then give a detailed exposition of susceptibilities, their empirical estimators, and their connection to the geometry of the loss landscape. |
| title | Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning |
| topic | Machine Learning Statistical Mechanics Statistics Theory |
| url | https://arxiv.org/abs/2605.07980 |